1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Alchen [17]
3 years ago
14

Which of the following is not one of the 8th roots of unity?

Mathematics
1 answer:
Anika [276]3 years ago
5 0

Answer:

1+i

Step-by-step explanation:

To find the 8th roots of unity, you have to find the trigonometric form of unity.

1.  Since z=1=1+0\cdot i, then

Rez=1,\\ \\Im z=0

and

|z|=\sqrt{1^2+0^2}=1,\\ \\\\\cos\varphi =\dfrac{Rez}{|z|}=\dfrac{1}{1}=1,\\ \\\sin\varphi =\dfrac{Imz}{|z|}=\dfrac{0}{1}=0.

This gives you \varphi=0.

Thus,

z=1\cdot(\cos 0+i\sin 0).

2. The 8th roots can be calculated using following formula:

\sqrt[8]{z}=\{\sqrt[8]{|z|} (\cos\dfrac{\varphi+2\pi k}{8}+i\sin \dfrac{\varphi+2\pi k}{8}), k=0,\ 1,\dots,7\}.

Now

at k=0,  z_0=\sqrt[8]{1} (\cos\dfrac{0+2\pi \cdot 0}{8}+i\sin \dfrac{0+2\pi \cdot 0}{8})=1\cdot (1+0\cdot i)=1;

at k=1,  z_1=\sqrt[8]{1} (\cos\dfrac{0+2\pi \cdot 1}{8}+i\sin \dfrac{0+2\pi \cdot 1}{8})=1\cdot (\dfrac{\sqrt{2}}{2}+i\dfrac{\sqrt{2}}{2})=\dfrac{\sqrt{2}}{2}+i\dfrac{\sqrt{2}}{2};

at k=2,  z_2=\sqrt[8]{1} (\cos\dfrac{0+2\pi \cdot 2}{8}+i\sin \dfrac{0+2\pi \cdot 2}{8})=1\cdot (0+1\cdot i)=i;

at k=3,  z_3=\sqrt[8]{1} (\cos\dfrac{0+2\pi \cdot 3}{8}+i\sin \dfrac{0+2\pi \cdot 3}{8})=1\cdot (-\dfrac{\sqrt{2}}{2}+i\dfrac{\sqrt{2}}{2})=-\dfrac{\sqrt{2}}{2}+i\dfrac{\sqrt{2}}{2};

at k=4,  z_4=\sqrt[8]{1} (\cos\dfrac{0+2\pi \cdot 4}{8}+i\sin \dfrac{0+2\pi \cdot 4}{8})=1\cdot (-1+0\cdot i)=-1;

at k=5,  z_5=\sqrt[8]{1} (\cos\dfrac{0+2\pi \cdot 5}{8}+i\sin \dfrac{0+2\pi \cdot 5}{8})=1\cdot (-\dfrac{\sqrt{2}}{2}-i\dfrac{\sqrt{2}}{2})=-\dfrac{\sqrt{2}}{2}-i\dfrac{\sqrt{2}}{2};

at k=6,  z_6=\sqrt[8]{1} (\cos\dfrac{0+2\pi \cdot 6}{8}+i\sin \dfrac{0+2\pi \cdot 6}{8})=1\cdot (0-1\cdot i)=-i;

at k=7,  z_7=\sqrt[8]{1} (\cos\dfrac{0+2\pi \cdot 7}{8}+i\sin \dfrac{0+2\pi \cdot 7}{8})=1\cdot (\dfrac{\sqrt{2}}{2}-i\dfrac{\sqrt{2}}{2})=\dfrac{\sqrt{2}}{2}-i\dfrac{\sqrt{2}}{2};

The 8th roots are

\{1,\ \dfrac{\sqrt{2}}{2}+i\dfrac{\sqrt{2}}{2},\ i, -\dfrac{\sqrt{2}}{2}+i\dfrac{\sqrt{2}}{2},\ -1, -\dfrac{\sqrt{2}}{2}-i\dfrac{\sqrt{2}}{2},\ -i,\ \dfrac{\sqrt{2}}{2}-i\dfrac{\sqrt{2}}{2}\}.

Option C is icncorrect.

You might be interested in
Write an equation in​ slope-intercept form of the line that passes through the given point and is perpendicular to the graph of
Keith_Richards [23]

Answer:

y = -5x - 17

Step-by-step explanation:

Linear equations, or lines, that are perpendicular to each other have opposite reciprocals for the value of slope.  For the given line, y = 1/5x - 2, the slope is 1/5 and the opposite reciprocal is -5.

Given the value of the slope and a given point on the second line, you can solve for 'b':

y = mx + b

8 = (-5)(-5) + b

8 = 25 + b

-17 = b

y = -5x - 17

3 0
3 years ago
Select the quadratic that has roots x=8 and x=-5 a x^2 + 3x -40
sergiy2304 [10]
If x = 8, then the factor is (x - 8); likewise, if x = -5, then the factor is (x + 5).  FOIL those together and get B
4 0
3 years ago
Jackie drinks a bottle of water and a bottle of orange juice every morning.
melisa1 [442]

Answer:

15.30

Step-by-step explanation:

use code tiko

3 0
3 years ago
Read 2 more answers
The area of a rectangle or a 48 square inches the length is 8 inches what is the measure of its width
adoni [48]
6 inches because area= length times width and 6 times 8= 48 inches
5 0
3 years ago
Read 2 more answers
Follow the steps to solve this equation. 3(2x + 6) – 4x = 2(5x – 2) + 6 1. Use the distributive property: 6x + 18 – 4x = 10x – 4
Yuliya22 [10]

Answer:

x=2.

Step-by-step explanation:

We have been given an equation 3(2x+6)-4x=2(5x-2)+6. We are supposed to solve our given equation step by step.

Step 1: Use the distributive property.

3\cdot 2x+3\cdot 6-4x=2\cdot 5x-2\cdot 2+6

6x+18-4x=10x-4+6

Step 2: Combine like terms.

6x-4x+18=10x-4+6

2x+18=10x+2

Step 3: Use subtraction property of equality:

2x-2x+18-2=10x-2x+2-2

16=8x

Step 4: Use the division property of equality:

\frac{16}{8}=\frac{8x}{8}

2=x

Therefore, the solution of our given equation is x=2.

8 0
3 years ago
Read 2 more answers
Other questions:
  • PLEASE HELP PLEASE HELP
    6·1 answer
  • Which of the following shows the extraneous solution to the logarithmic equation below???
    10·2 answers
  • Is there solutions for y<4x-6 and y>4x 2
    6·1 answer
  • A quadratic equation of the form 0=ax^2+bx+c has a discriminant value of -16. How many real number solutions does the equation h
    15·1 answer
  • 7 more than 3 times a number is 52
    8·1 answer
  • HELP ASAP PLEASE! 30 POINTS GIVEN
    11·2 answers
  • What is the greatest common factor of 48 and 64
    8·1 answer
  • Charles deposited $900 in a new account that earns 6% simple interest. After 2 years, how much interest will she have earned?
    8·1 answer
  • You are going on a road trip over a distance of 3000 miles with three friends. The car consumes 6 gallons of gas per 150 miles a
    9·1 answer
  • Given the function f(x) = x + 1 and the Linearr function g(x) which function has a greater value when x = 2
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!